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(a) Find a closed form expression for your marginal utility of happiness; that is, find a formula for M (w) which does not contain a limit. (b) Find a closed form for C'(w) and confirm that this number is always negative when w > 0, hence showing that the marginal utility of wealth is diminishing.

(a) Find a closed form expression for your marginal utility of happiness; that is, find a formula for M (w) which does not contain a limit. (b) Find a closed form for C'(w) and confirm that this number is always negative when w > 0, hence showing that the marginal utility of wealth is diminishing.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In economics, utility is a measure of satisfaction or usefulness. The utility of wealth could therefore be thought of as how happy you are as a function of the amount of wealth you possess. When applied to economic measurements, the word "marginal" describes the change corresponding to an increase in the measured quantity. The marginal utility of wealth is how much your happiness increases as your wealth increases. The Law of Diminishing Marginal Utility states that as you consume more, your marginal utility decreases; that is, the more you have of something, the smaller the impact getting more of it affects your happiness. If w is your wealth, and T(w) is your utility of wealth, your marginal utility of wealth at w is M(w) = lim h→0 T(w+h)-T(w) h and the rate at which your marginal utility of wealth is changing at w is M (w+h) — M(w) h C(w) = lim h→0 Suppose w is measured in 10s of thousands of dollars, and T(w) wealth. - 300w w + 1 is the utility of your (a) Find a closed form expression for your marginal utility of happiness; that is, find a formula for M(w) which does not contain a limit. (b) Find a closed form for C(w) and confirm that this number is always negative when w > 0, hence showing that the marginal utility of wealth is diminishing.
Transcribed Image Text:In economics, utility is a measure of satisfaction or usefulness. The utility of wealth could therefore be thought of as how happy you are as a function of the amount of wealth you possess. When applied to economic measurements, the word "marginal" describes the change corresponding to an increase in the measured quantity. The marginal utility of wealth is how much your happiness increases as your wealth increases. The Law of Diminishing Marginal Utility states that as you consume more, your marginal utility decreases; that is, the more you have of something, the smaller the impact getting more of it affects your happiness. If w is your wealth, and T(w) is your utility of wealth, your marginal utility of wealth at w is M(w) = lim h→0 T(w+h)-T(w) h and the rate at which your marginal utility of wealth is changing at w is M (w+h) — M(w) h C(w) = lim h→0 Suppose w is measured in 10s of thousands of dollars, and T(w) wealth. - 300w w + 1 is the utility of your (a) Find a closed form expression for your marginal utility of happiness; that is, find a formula for M(w) which does not contain a limit. (b) Find a closed form for C(w) and confirm that this number is always negative when w > 0, hence showing that the marginal utility of wealth is diminishing.
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